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Theorem cnvi 4671
Description: The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvi I = I

Proof of Theorem cnvi
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 x V
21ideq 4431 . . . 4 (y I xy = x)
3 equcom 1590 . . . 4 (y = xx = y)
42, 3bitri 173 . . 3 (y I xx = y)
54opabbii 3815 . 2 {⟨x, y⟩ ∣ y I x} = {⟨x, y⟩ ∣ x = y}
6 df-cnv 4296 . 2 I = {⟨x, y⟩ ∣ y I x}
7 df-id 4021 . 2 I = {⟨x, y⟩ ∣ x = y}
85, 6, 73eqtr4i 2067 1 I = I
Colors of variables: wff set class
Syntax hints:   = wceq 1242   class class class wbr 3755  {copab 3808   I cid 4016  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  coi2  4780  funi  4875  cnvresid  4916  fcoi1  5013  ssdomg  6194
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